On a compact smooth toric Kähler manifold \((M, J, \omega _0, \)  with Delzant polytope \(({\mathrm {P}}, {{\mathrm {L}}})\) ,
Donaldson [1]} found concise expressions for the Futaki invariant and the Mabuchi energy in terms of \(({\mathrm {P}}, {\mathrm {L}})\)  and the space \({\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\) . We use this point of view in order to express the extended Mabuchi energy as a functional acting on the data \((u, A, B)\)  associated to \(({\mathrm {P}}, {\mathrm {L}})\) . Following [1]}, we introduce a linear functional defined on the space of continuous functions on \({\mathrm {P}}\)  by
\({\bf F} (f) := - a\int _{{\mathrm {P}}} f(x){\mathrmsl d}x + 2 \int _{\partial {\mathrm {P}}} f(x) {\mathrmsl d}\sigma _{{\mathrm {L}}}, \qquad a:= 2\frac{\operatorname{Vol}\left({\partial {\mathrm {P}}}, {\mathrmsl d}\sigma _{{\mathrm {L}}}\right)}{\operatorname{Vol}({\mathrm {P}}, {\mathrmsl d}x)},\) 
